Parameterization of surface temperature retrieval for remote sensing experiments

Acta Astronautica Vol. 6, pp. 1061-1072 Pergamon Press Ltd., 1979. Printed in Great Britain

Parameterization of surface temperature retrieval for remote sensing experimentsT

S. N. T I W A R I ~ AND S. K. G U P T A §

Department of Mechanical Engineering and Mechanics, Old Dominion University, Norfolk, VA 23508, U.S.A.

Abstraet--A simple parameterization has been developed for determining the actual surface temperature from the effective brightness temperature measured radiometricaily in the 11 ~m window region. This algorithm allows the computation of atmospheric correction without performing detailed radiative transfer calculations. Correction due to atmospheric water vapor is represented in terms of the integrated water vapor burden. Correction due to variation of surface emittance is represented in terms of its deviation from unity. Parameteric representation has also been developed for simultaneous variation of both parameters. The parameterization is based on model calculations performed with a line-by-line radiative transfer program. Sensitivity of the retrieved surface temperature to uncertainties of water vapor burden and surface emittance have also been examined.

Introduction

MONITOmNO of atmospheric pollutants on large scale in space and time has become very important because of their considerable influence on human health, environment and meteorology. Further, remote sensing seems to be the only feasible method for measuring these pollutants on the desired scales. Several remote sensing instruments have been developed in recent years (Ward and Zwick, 1975; Reichle and Hesketh, 1976), designed to measure pollutant concentrations from aircraft, but these are readily adaptable for satellite use.

Extraction of pollutant concentrations from the signals obtained from these instruments requires prior knowledge of the temperature of the underlying surface, in addition to several other atmospheric parameters. Conventionally, broad band radiometers operating in the 11/zm window region have been used for measuring surface temperature. It has been found, however, that attenuation due to weak water vapor lines and the continuum in this spectral region is sufficiently important (Anding and Kauth, 1970; Prabhakara et al., 1974) so as to require correction for the radiometrically measured temperature. An additional correction is required because the emittance of most underlying surfaces is less than unity (Buettner and Kern, 1965; Kornfield and Susskind, 1977).

The water vapor attenuation can be satisfactorily accounted for using the

TPaper presented at the XXIXth Congress of the International Astronautical Federation, Dubrovnik, Yugoslavia, I-8 October 1978.

~Professor. §Research Associate.

1061

1062 S .N. Tiwari and S. K. Gupta

differential absorption method (Anding and Kauth, 1970). It requires either a two-channel or a dispersive instrument both of which are far more expensive than the broad-band radiometer. Alternatively, the correction can be evaluated by performing line-by-line radiative transfer calculations for a given water vapor distribution. However, the cost of such computations on a routine basis can be enormous. The purpose of the present study is to develop a simple parameterization based on line-by-line model calculations which permits evalua- tion of the corrections accurately and economically.

Theory and algorithm The quantity measured by a broad-band radiometer is the integrated up-

welling radiance which for a homogeneous non-scattering atmosphere can be expressed as

f~,i 2 Eo(~o) E = f(~o) d~o (1)

where to[ and oJ2 are the frequency bounds of the observed spectral region and f(oJ) is the instrument filter function. Eo(oJ) represents the monochromatic thermal radiance emitted by the underlying surface and the atmosphere and is given by

fo Eo(oo) = ~(oJ)B(oo, T~)'r(~o, 0) + B(~o, T(Z)) [d¢(oJ, Z)/dZI dZ (2)

where e(to) is the surface emittance, B(oJ, T) is the Planck's blackbody radiance, Ts is the surface temperature, T(Z) is the temperature at altitude Z, and r(~o, Z) is the monochromatic transmittance of the atmosphere. Solar radiation reflected from the surface and the scattered radiation make negligible contribution to the upwelling radiance in the 11/~m region.

The quantity measured by a direct-reading radiation thermometer is the effective brightness temperature (EBT) of the surface. It will be denoted by Te and is defined as

f ~B(¢o, T~)f(to) doJ = E (3) 1

where the blackbody radiance corresponding to 7", equals the total upwelling radiance. Under certain ideal conditions, namely (i) when the emittance of the target surface is unity, and (ii) when the attenuation by the atmosphere between the surface and the sensor is negligible, Te equals the actual surface temperature Ts. For nonideal conditions, however, correction has to be applied to measured Te to obtain 7",.

Analytical formulation The correction in the present work is defined as

A T = Te- Ts (4)

Parameterization of surface temperature retrieval for remote sensing experiments 1063

where T, is the measured value of EBT. Measurements of T, cannot be made, however, for all desired conditions necessary to establish the parametric rela- tions. They are obtained, therefore, by inverting eqn (3), where E is computed using a line-by-line radiative transfer program and the disired input data. Since T, is equal to Ts for surface emittance equal to unity and now water vapor in the atmosphere, these are adopted as the initial conditions. The quantities which give rise to correction AT, are the magnitude of the water vapor burden and deviation of surface emittance from unity, denoted by Aw and A~ respectively.

An analysis of the magnitudes of corrections A T caused by deviations Ax of a parameter shows the existence of a simple analytical relation between the two quantities. For most cases the relationship may be expressed by a polynomial as

N

AT = ~ a,,fAx)" (5) nffi0

where N is a positive integer. The first few terms of the polynomial adequately represent the correction AT for realistic values of deviations Ax. Model calculations were performed to evaluate AT for several values of Ax and the coefficents a, were obtained by polynomial regression. Results of these model calculations are discussed in the next section.

A more complicated situation arises when both parameters undergo simultaneous deviations. The combined correction, denoted by Ax and Ay represent the deviations of the two parameters x and y from their initial values, the combined correction may be expressed as

Arc = f ( A r x , Ary) (6)

where ATx and ATy are the individual corrections caused by deviations Ax and Ay separately. Additional calculations will be performed to establish the nature of the above functions and their results are also presented in the next section.

Water vapor absorption The total absorption coefficient, kt(aJ) for water vapor in the 11 ~m region

may be expressed as

kt(~o) = k,(co) + kp(~o)(p - Px2o) + k, Ox2o (7)

where p and px~o are the total pressure and partial pressure of water vapor respectively. The first term on the righthand side represents the line absorption and the next two terms represent the continuum absorption. Absorptions represented by k~ and k, are also called the p-type (foreign-broadened) and e-type (self-broadened) absorptions respectively. Bignell (1970) and Burch (1971) independently measured the values of kp and k, and showed that e-type absorption is the dominent process in this spectral region. Roberts et al. (1976) analyzed a large amount Of laboratory and field data on water vapor absorption in detail. A strong negative temperature dependence of k, was established by

1064 S .N. Tiwari and S. K. Gupta

these authors, again in agreement with the observations of Bignell (1970) and Burch (1971). It was shown that

k~(to ) = k~°(to ) exp [ T o ( l - ~ - ~ ) ] (8)

gives the value of k,(¢o) at the temperature, T, where k°(~o) refers to a reference temperature of 296 K. The best estimate of To was found to be 1800 K. The frequency dependence of k°(~o) was expressed as

k°(¢o) = a + b exp (-rico) (9)

where

a = 1.25 x 10 -22 mol -~ cm 2 atm -~

b = 2.34 x 10 -19 mol -~ cm 2 atm -l

/3 = 8 . 3 0 × 10 -3 cm.

The best estimate of kp is given by Selby et al. (1976). Spectral parameters for computing the line absorption were obtained from the AFCRL compilation (McClatchey et al., 1973).

Model atmosphere The model atmosphere used in the present work extends from the surface to

17,500 ft (approx. 5.3 kin). This is due to the fact that the work was performed to aid in reduction of data obtained from aircraft-borne instruments flying at midtropospheric altitudes. The pressure, temperature and water vapor profiles adopted here refer to the U.S. Standard Atmosphere (McClatchey et al., 1972). The atmosphere was divided into ten layers of unequal thickness with layer tops at 0.5, 1.5, 2.5, 6.5, 8.5, 10.5, 12.5, 14.5 and 17.5 lift (thousand feet) respectively. This choice of altitudes was dictated by the flying altitudes of the aircraft.

Model calculations and analysis The parameters whose variation was considered important for this study are

(i) water vapor burden, and (ii) surface emittance. Effects of the variation of atmospheric temperature distribution were also examined and were found to be much smaller. Model calculations have been performed and analyzed for divia- tion of each parameter individually and for both parameters simultaneously. The corrections for each value of the parameters are calculated for eight values of actual surface temperature, Ts and for ten altitudes, Z corresponding to the top of each layer.

Variation of water vapor distribution Variations of water vapor distribution were simulated by multiplying water

vapor concentration at each altitude by a constant factor. Computations of


correction to EBT were made for six multiples of the water vapor profile, namely, 0.0, 0.25, 0.5, 1.5, 2.0 and 3.0 in addition to the standard profile. The deviation of this parameter was expressed in units of the total water vapor burden corresponding to the standard distribution. Results for Z = 17,500 ft and 7", = 300, 310 and 320 K were chosen for illustration and are shown in Fig. 1. Polynomial regression in these results between Ax (water vapor burden) and AT (correction) shows that cubic expressions are required to adequately represent these relationships. Regression coefficients a0, a~, a2 and a3 were determined for all cases and a0 were found to be insignificantly small. Coefficients a~, a2 and a3 are found to be dependent on 7', as well as on the altitude. Figure 2 shows the variation of a~, a2 and a3 with 7", for Z = 17,500ft. Variation with Z for T, = 300 K is shown in Fig. 3.

The effect of redistribution of water vapor in the atmosphere (without changing the total water vapor burden) has also been studied by using two modifications of the standard water vapor profile. The first modification is obtained by arbitrarly decreasing the water vapor burden in the lower five layers and increasing the burden by the same amount in the upper five layers. The second modification was obtained by an exactly opposite operation. An examination of the results showed that the effect of these modifications on the magnitude of corrections is always smaller than 0.1 K and, therefore, was not considered significant.

Variation of surface emittance Variation of surface emittance was considered between 1.00 and 0.80 at

ALT • 17500 f t .

AT x • A 0+ A IL~x + A 2(L~x) 2 + A 3tAx) 3

= I 32O'K- / / g T,- 310"K \ / / _

WATER VAPOR BURDEN Fig. 1. V a x ~ t i o n o f cor rect ion w i th water vapor burden f r om 17,500 f t .

1066 S.N. Tiwari and S. K. Oupta

-0 .5 i l i ~ -0.4 --10.24 /

03 =1

-I.,S ---I.2 -~.16

=

-2.5 - -2.0 .08

-3 I I I -2 .8 .~ 290 ~ 310 320 330

/¢TUAL SURFACE TEM=ERA11.~, 1"1 (=K) Fig. 2. Variation of regression coemcients a=, a2 and a3 with actual surface temperature

for 17,500 ft.

0.4 I~ O.O 1012

O. -,~ a3 l - -0.2 .10

-- -(I.6 --~.06

-0.8 -~0.04

12 . . I I I I i .2 - - - ~ . o o 0 4 8 12 16 20

ALTITLI~ (kft) Fig. 3. Variation of regression coefficients a,, a2 and a3 with altitude for T, = 300 K.

intervals of 0.05. Computa t ions were made in presence of a s tandard water vapor burden in the a tmosphere for all eight values of T, and ten values of Z. It is important to note here that the total correct ion is due to at tenuation by water vapor as well as due to deviation of e. Correct ion due to deviation of • is obta ined by subtracting the correct ion which cor responds to • = 1.00 f rom the total and is shown in Fig. 4 for Z = 17,500ft and T, = 300, 310 and 3 2 0 K as a function of the deviat ion of emit tance, Ae. Polynomial regression in these data

Parameterization of surface tonpemture retrieval for remote sensing experiments 1067

-15

v o

- I0 i - r a uJ

n r o i t .

z o_ t - o.) i4J rr -5 ae O ¢.3

I

ALT = 175OO ft.

I 320" K T$, 310°K

300"K

AI x - A 0 +A!Ax ÷ A 2 fax) 2

I I

WET ATMOSPHERE

o V I I I I O.OO -0.05 -O.IO -O.15 -0.20

DEVIATION OF EMITTANCE

Fig. 4. Variation of correction with deviation o f surface emittance from 17,500 ft.

shows that these curves can be represented by quadratic expressions with residues much smaller than 0.I K. Regression coefficients a0, am and a2 were determined for all values of 7", and Z, and ao again were found to be very small for all cases. Coefficients am and a2 again are found to be dependent on 7', as well as Z. For illustration, the variation of am and a2 with 7", for Z = 17,500 ft is shown in Fig. 5. Figure 6 shows the variation of these coefficients with Z for

75 -15 I 1 I

" ~ ' ~ - ~ - W E T .,_ -2o • 01 ATMOSPHE ~ ' " ~

5 5 I I t -25 290 300 310 :320 330

ACTUAL SURFACE TEMPERATURE, TI PI0

Fig. 5. Variation of regression coefficients al and a2 with actual surface temperature for 1 7 J 0 0 ft.

1068 S.N. Tiwari and S. K. Gupta

6e / E ~ ~'a2 -19 z T s "300 ° K /

66 ~ -20

64 -2 I WET ATMOSPHERE

62 -- -22 Ol ~

6C I I J J 23 4 8 12 16 20 ALTITL[]E (kfl)

Fig. 6. Variation of the regression coefficients a~ and a2 with altitude for Ts = 300 K.

Ts = 300 K. It should be remembered here that the total correction will be obtained as

AT = AT, + ATw (10)

where AT, is the correction obtained using the above derived regression coefficients and ATw corresponds to the presence of standard water vapor burden.

Simultaneous variation of parameters It was realized that both parameters may vary at the same time and,

therefore, model calculations were performed to parameterize correction for simultaneous variation of the two parameters. Computations were made for the five values of • mentioned earlier and four values of water vapor burden expressed by 0.0, 0.5, 1.0 and 2.0. Attempts were made to represent the combined correction ATc in terms of the individual corrections AT, and AT, where the subscripts ~ and w refer to deviations of surface emittance and water vapor burden respectively. Several analytical representations were tried and it was found after some numerical experimentation that the combined correction can be expressed as

ATe = AT, + ATw + kAT, AT~. ( l l )

Value of the coefficient k was obtained for various combinations of the parameters Ae and Aw and was found to be constant. For example, for Ts = 323 K and Z = 17,500ft, a value of k =0.0197-+0.0003 was obtained. However, the value of k was found to vary with Ts as well as Z. Figure 7 shows the variation of k with Ts for Z = 17,500 ft (circles and solid lines) and also the variation with Z for 7", = 300 K (squares and dashed lines). Value of k for other than tabulated values of Z and T, can be obtained with reasonable accuracy by linear interpolation.


ACTUAL SURFACE TEMPERATURE, TsPK) 290 L:~ 300 305 310 315 520 525

0.00 I I I I I I

-o.oz k (Ts),Z = 17S00 FT ~ ~ . o . . _ . . . . o _ . . . . . _ ~

7 / - _ ~ ODO4

f _ . o - - - - o ' - - ' ° " . . . . J

-o= /

-o.o~ ~-//~'~" ~ - - k (z), T,-~O'K _

-O.IC I I I I 1 i 0.0 2.5 5.0 7.5 IOD 12.5 15.0 17.5

ALTITUDE, Z (kft)

Fig. 7. Variation of the co¢lficient k with actual surface temperature (solid curve) and with altitude (dashed curve).

Application of algorithm

It is evident from Figs. 2, 3 and 5-7 that regression coefficients a, and k are dependent on 7", as well as on the altitude Z. Figures 3, 6 and 7 further show that coefficients appropriate to any altitude between 500 and 17,500 ft can be obtained with desired accuracy using linear interpolation. Since 7", is the desired end result of the entire computation, an iterative procedure has been developed to arrive at its appropriate value.

Values of the regression coefficients a, and k for the appropriate altitude referring to T, = 300 K are adopted as their initial estimates. Correction is computed using these coefficients and is combined with EBT to obtain the first estimate, say T,t. Coefficients are then evaluated corresponding to T,, and the iteration is continued until successive estimates become consistent within 0.01 K.

Sample calculations A large number of sample calculations were made to check the algorithm for

various combinations of the deviations of both parameters. Table 1 shows a set of such results for a wide range of surface temperatures and altitudes. The sets of parameters chosen for these calculations were intended to show the largest possible differences between the actual and retrieved surface temperatures.

Table I. Results of sample calculations

AST(K) EBT(K) ALT(FT) DE [~4 ~T(K)

325 307.78 10,500 -0.20 0.00 325,00 315 296.48 12,500 -0.20 1.00 314.99 310 297.33 6,500 0.00 3.00 309,99 300 282.88 17,500 -0.20 2.00 300,10 320 295.36 17,500 -0.20 2.00 320,04

1070 S . N . Tiwari and S. K. G u p t a

These differences represent the cumulative error introduced in the result by the algorithm. The largest error observed here is only 0.1 K.

Sensitivity calculations It is important to examine the sensitivity of the retrieved surface temperature

to uncertainties of the two parameters considered in the present work. Sen- sitivity to uncertainties in the assumed water vapor burden has been estimated by computing EBT for water vapor burdens of 0.75 and 1.25 and comparing the results with EBT values obtained for the standard water vapor burden. The results presented in Fig. 8 show that a 25% variation of water vapor burden affects the measured EBT from 0.5 to 1.5 K for the range of Ts covered in the present work. It is inferred, therefore, that the surface temperature obtained from these values of EBT will have an average error of 1 K corresponding to a 25% uncertainty in the water vapor burden.

Sensitivity of the retrieved surface temperature to uncertainties in the assumed value of surface emittance may be estimated from the results of model calculations discussed earlier. Differences between EBT values referring to

= 1.0 and 0.95 are indicative of the differences between the measured values of EBT expected as a result of a 5% variation of surface emittance. Since surface temperature is obtained directly from EBT, it is assumed that these differences are typical of the errors introduced in the retrieval surface temperature due to 5% uncertainty in the assumed value of surface emittance. Figure 8 also shows

525 I I I

/

Noo t 0.75 ~ / / o OEN. 1,.oo 7"/

,., 3 , 5 - ( , .z5 . . . ~ " ~ . G /

,,, / / ' / ' / 0..

w / / / / ~ao,- //~//,//,'" I,d Z

" / / ' / " ,," ALT = . . . . . . . 17500 FT u.. IJJ / / ~ / / /

J / I I I 285

290 300 310 320 325

ACTUAL SURFACE TEMPERATURE (=K)

Fig. 8. Sensi t iv i ty o f E B T to uncer ta in ty in water vapor burden and surface emittance.

Parameterlzation of surface temperature retrieval for remote sensing experiments 1071

EBT as a function of Ts for Z = 17,500ft and e = 1.0 and 0.95. The average difference between the two lines is approx. 3.3 K. It is inferred, therefore, that surface emittance is an important parameter and great caution should be exercised in assuming its value.

Conclusions The present work shows that atmospheric correction to the effective bright-

ness temperature (EBT), as measured remotely by a radiation thermometer, can be evaluated using simple parameterizations. Analysis of model calculations shows that correction for variation of one parameter can be represented by simple analytical expressions in terms of a single variable. Correction for simultaneous variation of two parameters can be represented in terms of the individual corrections. It is shown further that these expressions can be used to evaluate corrections with high accuracy. Cumulative errors introduced by the algorithm do not exceed 0.1 K. Results of the sensitivity calculations show that the retrieved surface temperature is highly sensitive to uncertainties in assumed value of surface emittance. Sensitivity to uncertainties in water vapor burden is much smaller.

Acknowledgements--This work was supported by NASA Langley Research Center through Grant No. NSG-1282. The authors wish to express their gratitude to Dr. Henry G. Reichle, Jr. for several valuable discussions.

References Anding D. and Kauth R. (1970) Estimation of sea surface temperature from space. Remote Sensing

Environ. 1,217-220. Bignell K. J. (1970) The water vapor infrared continuum. Quarterly J. Roy. Met. Soc. 96, 390--403. Buettner K. J. K. and Kern C. D. (1965) The determination of infrared emissivities of terrestrial

surfaces. J. Geophys. Res. 70, 1329-1337. Butch D. E. (1971) Investigation of the absorption of infrared radiation by atmospheric bases.

Semi-annual Technical Report U-4784. Aeronutronic Division, Philco Ford Corp. Kornfield J. and Susskind J. (1977) On the effect of surface emissivity on temperature retrievals.

Mon. Wea. Rev. 105, 1605-1608. McClatchey R. A., Fenn R. W., Selby J. E. A., Volz F. E. and Garing J. S. (1972) Optical properties

of the atmosphere. AFCRL-72-0497, Air Force Cambridge Research Laboratories, Bedford, Mass.

McClatchey R. A., Benedict W. S., Clough S. A., Burch D. E., Calfee R. A., Fox K., Rothman L. S. and Garing J. S. (1973) AFCRL atmospheric line parameters compilation. AFCRL-TR-73-O096, Air Force Cambridge Research Laboratories, Bedford, Mass.

Prabhakara C., Dalu G. and Kunde V. G. (1974) Estimation of sea surface temperature from remote sensing in the 11-13/~m window region. J. Geophys. Res. 79, 5039-5044.

Reichle H. G. and Hesketh W. D. (1976) A gas-filter correlation instrument for atmospheric trace constituent monitoring. Presented at the W.M.O. Conference on Atmospheric Pollution Measurement Techniques, Gottenburg, Sweden.

Roberts R. E., Seiby J. E. A. and Biberman L. M. (1976) Infrared continuum absorption by atmospheric water vapor in the 8-12 ~m window. Applied Optics 15, 2085-2090.

Selby J. E. A., Shettle E. P. and McClatchey R. A. (1976) Atmospheric transmittance from 0.25 to 28.5 ~tm: Supplement LOWTRAN 3B. AFGL-TR-76-0285. Air Force Cambridge Research Laboratories, Hanscom AFB, Mass.

Ward T. V. and Zwick H. I4. (1975) Gas cell correlation spectrometer: GASPEC. Applied Optics 14, 2896-2904.

1072 S.N. Tiwari and S. K. Gupta

Appendix

Nomenclature ALT altitude EBT AST actual surface temperature E~

B(o~, T) Planck's blackbody radiance k~(~o) DE deviation of surface emittance kp(W)

DW deviation of water vapor burden k,(~0) E integrated upwelling radiance RST

effective brightness temperature monochromatic thermal radiance e-type absorption coeRicient for water vapor p-type absorption coefficient for water vapo~ total absorption coefficient for water vapor retrieved surface temperature

Documents

Parameterization of surface temperature retrieval for remote sensing experiments